Dynamic Energy Budget model for E. coli growth in carbon and nitrogen limitation conditions

Abstract The simulations and predictions obtained from mathematical models of bioprocesses conducted by microorganisms are not overvalued. Mechanistic models are bringing a better process understanding and the possibility of simulating unmeasurable variables. The Dynamic Energy Budget (DEB) model is an energy balance that can be formulated for any living organism and can be classified as a structured model. In this study, the DEB model was used to describe E. coli growth in a batch reactor in carbon and nitrogen substrate limitation conditions. The DEB model provides a possibility to follow the changes in the microbes’ cells including their elemental composition and content of some important cell ingredients in different growth phases in substrate limitation conditions which makes it more informative compared to Monod’s model. The model can be used as an optimal choice between Monod-like models and flux-based approaches. Key points • The DEB model can be used to catch changes in elemental composition of E. coli • Bacteria batch culture growth phases can be explained by the DEB model • The DEB model is more informative compared to Monod’s based models Supplementary Information The online version contains supplementary material available at 10.1007/s00253-024-13245-9.


Stock solution of Escherichia coli culture
LB medium dedicated to Escherichia coli culturing was prepared and sterilized at 121°C, 1 bar per 20 minutes.A single colony of E. coli (PCM 2057, Polish Academy of Science, Poland) from the LB plate (containing 2% agar powder (Sigma Aldrich, CAS: 9002-18-0)) was transferred into 100 mL of LB.The flask was incubated at 37°C for 24 hours and shaken at 160 rpm on a rotary shaker (IKA KS 4000, Germany), which led to the formation of a high-density culture.The LB medium with bacteria cells was diluted by sterile glycerol (CAS: 56-81-5, Poch, Poland) in proportion 60:40 (v:v) and transferred to the sterile tubes.The tubes were stored at -20ºC.

Inoculum
The cells from the E. coli stock solution were transferred on an LB plate using an inoculation loop and then incubated at 37°C for 24 hours.After this time a single colony of E. coli from the LB plate was transferred into 100 mL of sterile LB medium and incubated at 37°C for the next 24 hours and shaken at 160 rpm on a rotary shaker (IKA KS 4000, Germany).The bacteria cultures were in a stationary phase when used for further experiments.
The basic concentrations of mineral salts and glucose are presented in Table S1, and modifications of the M9 medium are described in Table S2.

Culturing and sampling
To 100 mL of sterile appropriate modified M9 medium 1% (1 mL) of inoculum was introduced.
The cultures were incubated at 37°C with shaking at 160 rpm on a rotary shaker (IKA KS 4000, Germany).The samples were collected in triplication while the culture was being grown.The concentrations of bacteria cells (optical density at 500 nm wavelength measurement), glucose, and nitrogen (colorimetric, spectrophotometric measurement) were measured.

Cells concentration analysis
The cell concentration was determined using the optical density method (OD).OD was measured at 550 nm (Shimadzu, USA) and recalculated into cell concentration using the standard curve given by the equation:  The standard curve was prepared by the cell suspension method.The cells suspension in basic M9 medium obtained after 24 hours of culturing was centrifuged (Hettich Unversal 320R, 20 min., 9000 rpm), the supernatant was discharged and the cells (precipitate) was washed by physiological salt solution (0.9% NaCl) and again centrifuged.After decantation, the cells were placed in the laboratory dryer (Memmert, 45°C) for 24h.Different amounts of dry cells were suspended in distilled water, and the absorbance at 550 nm wavelength was measured.Each sample was prepared in three repetitions.

Glucose analysis
Glucose concentration was determined with the commercial colorimetric enzymatic test (Biomaxima, Poland).The intensity of the color was measured spectrophotometrically at 500 nm.The standard curve was given by the equation: where: Cglucose -is a concentration of glucose [gL -1 ]; A500 -is an absorbance at 500 nm [-].The standard curve is shown in Figure S2.

Figure S2. Standard curve for glucose concentration
The cell suspension samples (see 1.4.Culturing and sampling) were centrifuged (3min, 6000 RPM).50 µL of supernatant was added to 1 mL of analytical enzymatic reagent (Biomaxima, Poland) and incubated for 5 min at 37 °C.After this time the absorbance at 500 nm was measured.
The color intensity measured photometrically is proportional to glucose concentration.The wavelength used to determine the concentration of glucose was selected on the spectrum of absorbance values of a solution containing 0.4 gL -1 of glucose, depending on the applied wavelength.The highest absorbance value was obtained for the wavelength of 500 nm (Figure S3).The intensity of the color was measured spectrophotometrically at 707 nm.The standard curve was given by the equation: where: CNH4Cl -is a concentration of ammonia chloride [gL -1 ]; A -is an absorbance at 707nm [-].The background A707 was measured, and equal to 0.057, and subtracted from further measurements.The standard curve is shown in Figure S4.
100 µL of supernatant was added to 5 mL of analytical 1 st reagent (NH4-1, Spectroquant, Merck, Germany).To the mixture 1 level blue microspoon (in the cap of the NH4-2 bootle -2 nd reagent) was added, mixed, and incubated for 10 min at room temperature.After this time the absorbance at 707 nm was measured.
This method determines a concentration of nitrogen in the form of ammonium nitrogen (NH4-N), ammonium (NH4 + ), ammonia nitrogen (NH3-N), and ammonia (NH3).In a strong alkaline solution ammonium nitrogen is present almost entirely as ammonia, which reacts with hypochlorite ions to form monochloramine.This in turn reacts with a substituted phenol to form a blue indophenol derivative that is determined photometrically.Due to the intrinsic yellow coloration of the reagent blank, the measurement solution is yellow-green to green color.
The wavelength used to determine the concentration of NH4Cl was selected on the spectrum of absorbance values of a solution containing 0.4 gL -1 of NH4Cl, depending on the applied wavelength.
The highest absorbance value was obtained for the wavelength of 707 nm (Figure S5).

Biomass composition
The content of C, H, N, and S in dry bacterial pellets obtained from C-limited and N-limited cultures was determined by CHNS Elemental Analyzer Vario EL Cube, with acetanilide as a standard (in the external laboratory).The results are given in Table S4.where:  denotes the residual with unknown molecular weight.Probably the main component of both residuals is oxygen.Slightly lower  and  stoichiometric coefficients of E. coli biomass were determined previously for ammonium-limited and glucose-limited growth in a continuous reactor (Folsom and Carlson, 2015).
The molecular weight of C-limited and N-limited biomasses can be easily calculated as a ratio between sample mass and molar content of C in the sample.Therefore the mean molar weights of C-limited, and N-limited biomasses are equal  BClim = 30.88[g mol -1 ] and  BNlim = 29.11[g mol -1 ] respectively.
According to the DEB theory it can be assumed that biomass is built only from three components structure, C-reserve and N-reserve.Therefore, the biomass composition can be expressed as: where       denotes  C-moles of structures with unknown stoichiometric coefficients ,  and ;  2  - C-moles of C-reserve which has the same elemental composition as glucose;  3 - moles of N-reserve with the composition the same as ammonia;      one C-mole of biomass in certain conditions with measured  and  values.Note that equation (S4) does not describe a chemical reaction, it is used to split biomass into three different components.
It can be assumed that the content of C-reserve in C-limited bacterial culture is equal to zero ( = 0), as well as N-reserve content in N-limited cultures ( = 0).Therefore, the composition of C-limited and Nlimited biomass can be expressed as: Note that the sulfur content was negligibly small and therefore it can be omitted, and the oxygen content hidden in residuals does not have to be known in further considerations.The mass balance for C, H, and N for C-limited biomass is given by the set of equations: and for N-limited biomass: The composition of structures is assumed to be constant.The set of six equations (S7) -(S12) can be simplified given that  1 = 1 to five equations with five unknowns  2 , , , , and .This set of nonlinear equations was solved using the Trust-Region-Dogleg algorithm in Matlab.The results were:  2 = 0.99,  = 0.0149,  = 0.027,  = 1.96, and  = 0.22.Therefore the composition C-mol of structures can be expressed as  1.96    0.22 .The stoichiometric coefficient  was assumed to be equal to 0.45 based on the (Folsom and Carlson, 2015) and used only in material balance calculations.The molecular weight of C-mol of structure  Mv can be calculated from the equations: and/or where   , and   are molecular weight of C-moles of C-, and N-reserves equal to the molecular weight of C-mol of glucose and ammonium ion, 30.0267 and 18 [g mol -1 ] respectively.
The mean value of molecular weight of C-mol of structure  Mv was equal to 29.76 [g mol -1 ], and was used along with   and   for mass-moles recalculation within the DEB model.The mean biomass C-mol molecular weight was equal to   = 30.00[g mol -1 ] and was used within Monod's model.
It should be noted that the solution of six equations (S7) -( S12) is highly sensitive to the stoichiometric coefficients of C-mol of biomass (S5) and (S6).Therefore the results can be fraught with considerable error.

Shock limitation and reserves kinetics
The shock limitation experiment was designed to investigate the reserve density kinetics and to verify the growth and maintenance constraints.The growing cells were transferred to the medium where one of substrates, glucose or ammonia, was missing and therefore could not be used to supply reserves.The model predictions were compared with measured values.The simulation shows the reserves, structure, and biomass dynamics before, and right after the growing cells were transferred to the medium without C-substrate (raw A) or N-substrate (raw B) (Fig. S6).In case when the bacteria cells were transferred to the medium without glucose, the model predicts a decrease in biomass due to the decrease in C-reserve density and gentle increase in structures (Fig. S6).
In the case when bacteria were transferred into a medium without ammonia small increase in biomass and C-reserves was predicted.The growth and maintenance can continue until the N-reserve is sufficiently filled.N-reserve density is low and therefore it has a low influence on total biomass.
Nevertheless, the model prediction and constraints used were confirmed only by the data in time intervals where the biomass does not change anymore (Fig. S6).

Mass balance
The material balance for each of the four main elements C, H, O, and N is given by flowing equation (S15): where   is a vector of total 'organic' fluxes S1) where: CE.coli -is a concentration of dry E. coli biomass [gL -1 ]; A550 -is an absorbance at 550 nm [-].The background absorbance was measured (A550 = 0.014) and subtracted from further measurements.The standard curve is shown in Figure S1.

Figure S1 .
Figure S1.Standard curve for cell dry biomass

Figure S3 .
Figure S3.The spectrum of the absorbance value of the solution containing 0.4 g L -1 glucose depending on the wavelength used

Figure
Figure S5.The spectrum of the absorbance value of the solution containing 0.1 g L -1 NH4Cl depending on the wavelength used

Figure S6 .
Figure S6.DEB and Monod's model simulation to experimental data obtained in shock limitation experiments.The simulation shows the reserves, structure, and biomass dynamics before and after the growing cells were transferred to the medium without C-substrate (raw A) or N-substrate (raw B).Model parameter values listed in Tab. 1 were used.
this part with the main model equations(10 -12).  is the stoichiometric coefficients matrix for 'organic' substances: is a vector of minerals (or metabolites) fluxes [mol h -1 ]: S15) can be solved to calculate the total mineral fluxes:  = −  −     (S22)The equation (S22) was solved numerically with model equations (10 -12) for estimated parameter values giving the total produced amounts of metabolites (per liter of growth medium) in time.An example of metabolites production/consumption analysis was prepared for the C-limitation scenario with an initial glucose concentration of 1 gL-1 and is presented in FigureS7.

Figure S7 .
Figure S7.Amounts of metabolites produced in time.

Figure S8 .
Figure S8.Amounts of metabolites produced in time.

Table S4 . The mass content of C, H, N, and S in samples of bacterial pellet obtained in C-limited and N-limited growth conditions.
The content of each element in the sample can be recalculated for given molecular weights and expressed as a mean number of moles per one mole of carbon (C-mole).Therefore the C-mole of C-limited biomass 1       +  3 =  2.04  0.25   (S5)  2       +  2  =  1.96  0.22   (S6) (Kooijman, 2010)ter J, * stands for substrates   ,   , reserves,   ,   or structure ) [mol h-1].The name 'organic' is used in the DEB book(Kooijman, 2010), however, the set of fluxes in vector   describes general changes of variables used in the model: substates, reserves, and structures (no matter if they are actually organic or not, like ammonia).Note that reserves are expressed in absolute values    [mol h -1 ], not as reserves densities, therefore:     = (    −     +        )  (S17)Vector   can be expressed using specific fluxes ̇ * [mol C-molMV -1 h -1 ] as follows: